Besides, several strategies have been exploited for the ill-conditioned Toeplitz systems, such as \(\tau \) -preconditioners (which can be diagonalized by the discrete sine transform matrix) [1]}, [2]}, [3]}, [4]}, circulant preconditioners [5]}, [6]}, [7]}, [8]}, and multigrid methods [9]}, [10]}, [11]}, [12]}, [13]}, [14]}. These approaches can significantly speed up the convergence
of iterative methods for solving Toeplitz systems where their
generating functions have an
\({\alpha }\) -th order zero at the origin (\(1 < \alpha < 2\) ).
However, the linear convergence of these methods
cannot be theoretically confirmed
for solving such Toeplitz systems.
Lately, some efficient preconditioners are developed to solve
linear systems arising from fractional diffusion equations;
see [15]}, [16]}, [17]}, [18]}, [19]}. Nevertheless, from the theoretical point of view, the spectra of these preconditioned matrices are not shown to
be bounded independent of the matrix sizes
and hence the linear convergence cannot be guaranteed.
